Learning Functional Graphs with Nonlinear Sufficient Dimension Reduction

Technical Summary: Learning Functional Graphs with Nonlinear Sufficient Dimension Reduction

Problem Statement
Functional graphical models are designed to estimate network structures where observations at vertices are random functions rather than scalar variables. This is particularly relevant in fields like neuroimaging (e.g., f-MRI, EEG), where data consists of time-series signals across brain regions. Existing methods for functional graphical models face significant limitations:

1. Parametric Assumptions: Methods like the Functional Gaussian Graphical Model (FGGM) and Functional Copula Gaussian Graphical Model rely on Gaussian or copula-Gaussian assumptions, which may not hold for complex real-world data.
2. Additivity Constraints: Nonparametric approaches based on Additive Conditional Independence (ACI) relax distributional assumptions but impose an additive structure on the relationships, which may not capture general nonlinear dependencies.
3. Curse of Dimensionality: Directly assessing conditional independence among high-dimensional functional vectors ($X_{-(i,j)}$) is hindered by the curse of dimensionality, leading to poor estimation accuracy.

Methodology: The Functional Sufficient Graphical Model (f-SGM)
The authors propose a nonparametric functional graphical model based on nonlinear Sufficient Dimension Reduction (SDR). The method, termed f-SGM, constructs the graph through two primary steps at both the population and sample levels:

1. Functional Nonlinear SDR (Dimension Reduction):

   • For each pair of nodes $(i, j)$, the method treats the pair $(X_i, X_j)$ as the response and the remaining nodes $X_{-(i,j)}$ as predictors.
   • It employs Functional Generalized Sliced Inverse Regression (f-GSIR) to find a low-dimensional sufficient predictor, denoted as $U_{ij}$.
   • This step relies on Reproducing Kernel Hilbert Spaces (RKHS) and nested kernels to map the infinite-dimensional functional data into a finite-dimensional space. The central $\sigma$-field for predicting $(X_i, X_j)$ given $X_{-(i,j)}$ is identified as the range of a regression operator within this RKHS framework.
   • Crucially, this reduces the high-dimensional conditioning set $X_{-(i,j)}$ to a low-dimensional vector $U_{ij}$ such that $(X_i, X_j) \perp X_{-(i,j)} \mid U_{ij}$.

2. Conditional Independence Testing via Hybrid CCCO:

   • Once $U_{ij}$ is obtained, the method tests for conditional independence $X_i \perp X_j \mid U_{ij}$.
   • To handle the mixture of functional data ($X_i, X_j$ in Hilbert spaces) and the Euclidean sufficient predictor ($U_{ij}$), the authors introduce a Hybrid Conjoined Conditional Covariance Operator (Hybrid CCCO).
   • This operator is defined within a tensor product of RKHSs generated by kernels for the functional components and the Euclidean vector.
   • The absence of an edge between $i$ and $j$ is determined if the Hilbert-Schmidt norm of the Hybrid CCCO is zero (or below a threshold).

Implementation Details

• Sample Level: The authors develop algorithms using coordinate mappings to represent Hilbert space elements and operators as vectors and matrices.
• Kernels: They utilize the Brownian motion kernel for the first-level Hilbert spaces (approximating the functional data) and Gaussian Radial Basis Functions (RBF) for the second and third-level spaces (SDR and CCCO).
• Tuning: Tuning parameters for Tychonoff regularization and kernel bandwidths are selected using Generalized Cross-Validation (GCV). The edge threshold is also determined via GCV to minimize prediction error.

Key Contributions

1. Nonparametric Flexibility: Unlike FGGM or copula-based models, f-SGM does not require Gaussian or specific parametric distributional assumptions.
2. Beyond Additivity: Unlike ACI-based methods, f-SGM does not assume additive conditional independence, allowing it to capture general nonlinear dependencies.
3. Curse of Dimensionality Mitigation: By reducing the conditioning set to a sufficient predictor $U_{ij}$, the method avoids the high-dimensional estimation issues associated with fully nonparametric approaches.
4. Hybrid Operator Theory: The paper rigorously defines the Hybrid CCCO to handle the specific structural challenge of testing independence between Hilbertian random functions conditioned on a Euclidean vector.

Results
The paper evaluates f-SGM through simulation studies and a real-world application:

• Simulations: The method was compared against FGGM, the Functional Additive Precision Operator (FAPO), and a naive SGM (n-SGM) using principal components.
  • In nonlinear additive models, f-SGM performed comparably to FAPO and significantly better than FGGM and n-SGM.
  • In nonlinear models with heteroscedasticity (where dependence exists in conditional variance rather than mean), f-SGM significantly outperformed all other methods, including FAPO, which failed to capture non-additive variance structures.
  • In Gaussian models, while FGGM (the parametric oracle) performed best, f-SGM demonstrated superior performance over FAPO and n-SGM, indicating it loses less information than additive nonparametric methods even when the data is Gaussian.
  • The method remained robust as the dimension $p$ increased (up to $p=40$) and sample sizes varied.
• Application (f-MRI): The method was applied to an f-MRI dataset from the ADHD consortium (171 children: 73 ADHD, 98 control). The analysis aimed to detect differences in brain network structures between the two groups. The f-SGM results revealed distinct network differences between ADHD and control groups that were less apparent or structured differently in the networks estimated by FAPO, FGGM, and n-SGM.

Significance and Claims
The authors claim that f-SGM offers a flexible, nonparametric framework for learning functional graphical models that effectively balances the relaxation of distributional assumptions with high estimation accuracy. By integrating nonlinear SDR with conditional independence testing via a hybrid operator, the method addresses the "curse of dimensionality" without resorting to restrictive additivity assumptions. The paper positions f-SGM as a significant advancement for analyzing complex multivariate functional data, particularly in neuroscience, where capturing non-Gaussian and non-additive dependencies is critical.

The authors modestly note that while the methodology and theoretical structure at the population level are established, asymptotic properties such as estimation consistency, convergence rates, and statistical inference are left for future research.